Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $x = \dfrac{35p + 7}{6p} \div \dfrac{2(5p + 1)}{p} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{35p + 7}{6p} \times \dfrac{p}{2(5p + 1)} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ (35p + 7) \times p } { 6p \times 2(5p + 1) } $ $ x = \dfrac {p \times 7(5p + 1)} {6p \times 2(5p + 1)} $ $ x = \dfrac{7p(5p + 1)}{12p(5p + 1)} $ We can cancel the $5p + 1$ so long as $5p + 1 \neq 0$ Therefore $p \neq -\dfrac{1}{5}$ $x = \dfrac{7p \cancel{(5p + 1})}{12p \cancel{(5p + 1)}} = \dfrac{7p}{12p} = \dfrac{7}{12} $